A circle has a radius of $5$. An arc in this circle has a central angle of $\dfrac{82}{45}\pi$ radians. What is the length of the arc? ${10\pi}$ ${\dfrac{82}{45}\pi}$ $\color{#DF0030}{\dfrac{82}{9}\pi}$ ${5}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (5) = 10\pi$ The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{82}{45}\pi \div 2 \pi = \dfrac{s}{10\pi}$ $\dfrac{41}{45} = \dfrac{s}{10\pi}$ $\dfrac{41}{45} \times 10\pi = s$ $\dfrac{82}{9}\pi = s$